The concepts of limits form the cornerstone for the most important Calculus theories (integration and differentiation). As a matter of fact, these two fields in calculus are defined using the processes of limits. The emphasis given in studying limits is so important that one should never jump directly to derivatives and anti-derivatives (more commonly known as integrals) without gaining a firm understanding of limit concepts (consider this to be a building block of calculus). So let's begin...

What is a limit?

The word "limit" is a concept used to describe the value of a function as it "approaches" a certain point. For example, a car going to certain place must follow corresponding speed limits or else, the driver will be penalized. In that case, the function can be assigned to the speed of the car and the limit of that function is the value of the car's speed. Now, consider f as a function given by y=f(x) and suppose that f is defined at each point x along a determined open interval I which contains a. So when we say that "f(x) approaches L as x approaches a'" or "L is the limit of f(x), as x approaches a", this means that f(x) will get closer to L as x gets closer to a.To better understand this, take a look at the following examples.

Example #1

Let f be a function defined by y=f(x)=x and a=3. Observe that f is defined for all x (for every interval I containing a=3). Let us see and investigate the value of the function f(x) when x is close to zero but not equal to zero.

f(x)

5

4

3.9

3.6

3.5

3.1

3.01

3.0001

x

5

4

3.9

3.6

3.5

3.1

3.01

3.0001

Table 1: Values of f(x)=x for x>0

f(x)

-1

0

1

2

2.5

2.7

2.8

2.999

x

-1

0

1

2

2.5

2.7

2.8

2.999

Table 2: Values of f(x)=x for x<0

From Table 1 and Table 2 above, notice that the value of the function f(x) gets closer to 3 as x gets closer to 3. This happens because 3 is the limit of the function f(x) as x approaches to 3. Thus, the concept of a limit is something that pertains to the behaviour of a function, in this case f(x). We may write it in the following notation:

\lim_{x \to 3}x=3

Example #2

Let f be a function defined by y=f(x)=3x+1 and a=0. Observe that f is defined for all x (for every interval I containing a=0). Let us see and investigate the value of the function f(x) when x is close to zero but not equal to zero.

f(x)

4

2.5

1.75

1.3

1.03

1.003

1.0003

1.00003

x

1

0.5

0.25

0.10

0.01

0.001

0.0001

0. 00001

Table 3: Values of f(x)=3x+1 for x>0

f(x)

-2

-0.5

-0.25

-0.7

-0.97

-0.997

-0.9997

-0.99997

x

-1

-0.5

-0.25

-0.10

-0.01

-0.001

-0.0001

-0. 00001

Table 4: Values of f(x)=3x+1 for x<0

From Table 3 and Table 4 above, notice that the value of f(x)=3x+1 gets closer to 1 as x gets closer to zero. This happens because 1 is the limit of the function f(x) as x approaches to zero. Thus, the concept of a limit is something that pertains to the behaviour of a function, in this case f(x). We may write it in the following notation:

\lim_{x \to 0}3x+1=1

Did you notice that the limit of the function f(x)=3x+1 as x approaches zero is the same as the value of the function at zero (f(0)=1) itself? Well, you're right but not all the time. Why? Well, consider this...

Example #3

Let f be a function defined by f(x)=\frac{x^{2}-2x}{x-2}. Let us see how this function behaves as it approaches to the point a=2. That is, let us find the limit of f as it approaches a.

Solution: At x=2, the function is not defined because we will have zero at the denominator of f. However, the is function defined at each x in any open interval I containing a=2 (except at 2). How could this be? Well, let's look at the following table.

f(x)

3

2.5

2.1

2.01

2.001

2.0001

2.00001

2.000001

x

3

2.5

2.1

2.01

2.001

2.0001

2.00001

2.000001

Table 5: Values of f(x)=\frac{x^{2}-2x}{x-2} for x>0

f(x)

1

1.5

1.9

1.99

1.999

1.9999

1.99999

1.999999

x

-1

-1.5

-1.9

-1.99

-1.999

-1.9999

-1.99999

-1.999999

Table 6: Values of f(x)=\frac{x^{2}-2x}{x-2} for x<0

We see from Table 5 and Table 6 that the function f gets closer to 2 as x gets closer to 2. Again, this is expected because 2 is the limit of f(x) as x approaches 2,

\lim_{x \to 2}f(x) = \lim_{x \to 2}\frac{x^{2}-2x}{x-2}=2

Thus, we say that the limit of f as x approaches to a number a is L, it means that the value of f can be made closer to L as we please by taking x close enough to a.

Example #4

In example #3, we solved for the limit of f(x)=\frac{x^{2}-2x}{x-2} as it approaches the point 2 which is just 2. Let us see how this function behaves as it approaches another point which is the point 0. That is, let us find the limit of f as it approaches a=0.

Solution: At x=0, the function is well-defined, so we can proceed with our limits calculation. By looking at the tables below,

f(x)

3

2.5

2.0

1.5

1.2

1.0

0.5

.01

x

3

2.5

2.0

1.5

1.2

1.0

0.5

.01

Table 7: Values of f(x)=\frac{x^{2}-2x}{x-2} for x>0

f(x)

-3

-2.5

-2.0

-1.5

-1.2

-1.0

-0.5

-0.01

x

-3

-2.5

-2.0

-1.5

-1.2

-1.0

-0.5

-0.01

Table 8: Values of f(x)=\frac{x^{2}-2x}{x-2} for x<0

We see from Table 7 and Table 8 that the function f gets closer to 0 as x gets closer to 0. Again, this is expected because 0 is the limit of f(x) as x approaches 0,

\lim_{x \to 0}f(x) = \lim_{x \to 0}\frac{x^{2}-2x}{x-2}=0

Example #5

In the preceding examples, the functions involved are non-constant. So what happens if a function is a constant? Say f=c, where c is a constant. Then what would be the limit of f as it approaches a=0. If we're going to make a table of values,

f(x)

c

c

c

c

c

c

c

c

x

3

2.5

2.0

1.5

1.2

1.0

0.5

.01

Table 7: Values of f(x)=\frac{x^{2}-2x}{x-2} for x>0

f(x)

c

c

c

c

c

c

c

c

x

-3

-2.5

-2.0

-1.5

-1.2

-1.0

-0.5

-0.01

Table 8: Values of f(x)=\frac{x^{2}-2x}{x-2} for x<0

As seen from the table, whatever the values of x is, the value of the function remains unchanged. So it is safe to say that the limit of a constant is the constant itself. Thus we can say